Developments in Parrondo's Paradox

Abstract

Parrondo’s paradox is the well-known counterintuitive situation where individually losing strategies or deleterious effects can combine to win. In 1996, Parrondo’s games were devised illustrating this effect for the first time in a simple coin tossing scenario. It turns out that, by analogy, Parrondo’s original games are a discrete-time, discrete-space version of a flashing Brownian ratchet—this was later formally proven via discretization of the Fokker-Planck equation. Over the past ten years, a number of authors have pointed to the generality of Parrondian behavior, and many examples ranging from physics to population genetics have been reported. In its most general form, Parrondo’s paradox can occur where there is a nonlinear interaction of random behavior with an asymmetry, and can be mathematically understood in terms of a convex linear combination. Many effects, where randomness plays a constructive role, such as stochastic resonance, volatility pumping, the Brazil nut paradox etc., can all be viewed as being in the class of Parrondian phenomena. We will briefly review Parrondo’s paradox, its recent developments, and its connection to related phenomena. In particular, we will review in detail a new form of Parrondo’s paradox: the Allison mixture—this is where random sequences with zero autocorrelation can be randomly mixed, paradoxically producing a sequence with non-zero autocorrelation. The equations for the autocorrelation have been previously analytically derived, but, for the first time, we will now give a complete physical picture that explains this phenomenon.